# 5.1 Double integrals over rectangular regions  (Page 3/12)

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Since $\text{Δ}A=\text{Δ}x\text{Δ}y=\text{Δ}y\text{Δ}x,$ we can express $dA$ as $dx\phantom{\rule{0.2em}{0ex}}dy$ or $dy\phantom{\rule{0.2em}{0ex}}dx.$ This means that, when we are using rectangular coordinates, the double integral over a region $R$ denoted by $\underset{R}{\iint }f\left(x,y\right)dA$ can be written as $\underset{R}{\iint }f\left(x,y\right)dx\phantom{\rule{0.2em}{0ex}}dy$ or $\underset{R}{\iint }f\left(x,y\right)dy\phantom{\rule{0.2em}{0ex}}dx.$

Now let’s list some of the properties that can be helpful to compute double integrals.

## Properties of double integrals

The properties of double integrals are very helpful when computing them or otherwise working with them. We list here six properties of double integrals. Properties 1 and 2 are referred to as the linearity of the integral, property 3 is the additivity of the integral, property 4 is the monotonicity of the integral, and property 5 is used to find the bounds of the integral. Property 6 is used if $f\left(x,y\right)$ is a product of two functions $g\left(x\right)$ and $h\left(y\right).$

## Properties of double integrals

Assume that the functions $f\left(x,y\right)$ and $g\left(x,y\right)$ are integrable over the rectangular region R ; S and T are subregions of R ; and assume that m and M are real numbers.

1. The sum $f\left(x,y\right)+g\left(x,y\right)$ is integrable and
$\underset{R}{\iint }\left[f\left(x,y\right)+g\left(x,y\right)\right]dA=\underset{R}{\iint }f\left(x,y\right)dA+\underset{R}{\iint }g\left(x,y\right)dA.$
2. If c is a constant, then $cf\left(x,y\right)$ is integrable and
$\underset{R}{\iint }cf\left(x,y\right)dA=c\underset{R}{\iint }f\left(x,y\right)dA.$
3. If $R=S\cup T$ and $S\cap T=\varnothing$ except an overlap on the boundaries, then
$\underset{R}{\iint }f\left(x,y\right)dA=\underset{S}{\iint }f\left(x,y\right)dA+\underset{T}{\iint }f\left(x,y\right)dA.$
4. If $f\left(x,y\right)\ge g\left(x,y\right)$ for $\left(x,y\right)$ in $R,$ then
$\underset{R}{\iint }f\left(x,y\right)dA=\underset{R}{\iint }g\left(x,y\right)dA.$
5. If $m\le f\left(x,y\right)\le M,$ then
$m\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}A\left(R\right)\le \underset{R}{\iint }f\left(x,y\right)dA\le M\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}A\left(R\right).$
6. In the case where $f\left(x,y\right)$ can be factored as a product of a function $g\left(x\right)$ of $x$ only and a function $h\left(y\right)$ of $y$ only, then over the region $R=\left\{\left(x,y\right)|a\le x\le b,c\le y\le d\right\},$ the double integral can be written as
$\underset{R}{\iint }f\left(x,y\right)dA=\left({\int }_{a}^{b}g\left(x\right)dx\right)\left({\int }_{c}^{d}h\left(y\right)dy\right).$

These properties are used in the evaluation of double integrals, as we will see later. We will become skilled in using these properties once we become familiar with the computational tools of double integrals. So let’s get to that now.

## Iterated integrals

So far, we have seen how to set up a double integral and how to obtain an approximate value for it. We can also imagine that evaluating double integrals by using the definition can be a very lengthy process if we choose larger values for $m$ and $n.$ Therefore, we need a practical and convenient technique for computing double integrals. In other words, we need to learn how to compute double integrals without employing the definition that uses limits and double sums.

The basic idea is that the evaluation becomes easier if we can break a double integral into single integrals by integrating first with respect to one variable and then with respect to the other. The key tool we need is called an iterated integral.

## Definition

Assume $a,b,c,$ and $d$ are real numbers. We define an iterated integral    for a function $f\left(x,y\right)$ over the rectangular region $R$ $=\left[a,b\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[c,d\right]$ as

1. $\underset{a}{\overset{b}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{c}{\overset{d}{\int }}f\left(x,y\right)dy\phantom{\rule{0.2em}{0ex}}dx=\underset{a}{\overset{b}{\int }}\left[\underset{c}{\overset{d}{\int }}f\left(x,y\right)dy\right]dx$

2. $\underset{c}{\overset{d}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{a}{\overset{b}{\int }}f\left(x,y\right)dx\phantom{\rule{0.2em}{0ex}}dy=\underset{c}{\overset{d}{\int }}\left[\underset{a}{\overset{b}{\int }}f\left(x,y\right)dx\right]dy.$

The notation $\underset{a}{\overset{b}{\int }}\left[\underset{c}{\overset{d}{\int }}f\left(x,y\right)dy\right]dx$ means that we integrate $f\left(x,y\right)$ with respect to y while holding x constant. Similarly, the notation $\underset{c}{\overset{d}{\int }}\left[\underset{a}{\overset{b}{\int }}f\left(x,y\right)dx\right]dy$ means that we integrate $f\left(x,y\right)$ with respect to x while holding y constant. The fact that double integrals can be split into iterated integrals is expressed in Fubini’s theorem. Think of this theorem as an essential tool for evaluating double integrals.

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